There is a well-known circuit for controlling the electrical fuel pump in accordance with an rpm signal in order that, for example, in an accident and with a resultant sudden stopping of the internal combustion engine, no more fuel is supplied to the engine. In this known circuit, a capacitor is charged in accordance with an rpm signal and the value of the discharging time constant of the capacitor is kept constant. However, the specific timing condition established on the basis of certain electrical properties does not allow an optimum cut-off operation for the internal combustion engine.